Question: For some integers that are not palindromes, like 91, a person can create a palindrome by repeatedly reversing the number and adding the original number to its reverse. For example, $91 + 19 = 110$. Then $110+011 = 121$, which is a palindrome, so 91 takes two steps to become a palindrome. Of all positive integers between 10 and 100, what is the sum of the non-palindrome integers that take exactly six steps to become palindromes?
Answer: Say our two-digit non-palindrome is $n=\overline{ab}=10a+b$, with $a$ and $b$ digits. Reversing $n$ and adding it to itself is $10a+b+10b+a=11(a+b)$. This operation only depends on $a+b$, so 57 and 48 for example yield the same result. When $a+b\le9$, the resulting number is just a number in $\{11,22,\ldots,99\}$, all of which are palindromes, so numbers with $a+b\le9$ take one step. We can now check how many times the operation needs to be applied on each remaining value of $a+b$. Since $a,b\le9$, $a+b\le18$. \[
a+b=10 \rightarrow 110 \rightarrow 121
\] \[
a+b=11 \rightarrow 121
\] \[
a+b=12 \rightarrow 132 \rightarrow 363
\] \[
a+b=13 \rightarrow 143 \rightarrow 484
\] \[
a+b=14 \rightarrow 154 \rightarrow 605 \rightarrow 1111
\] \[
a+b=15 \rightarrow 165 \rightarrow 726 \rightarrow 1353 \rightarrow 4884
\] \[
a+b=16 \rightarrow 176 \rightarrow 847 \rightarrow 1595 \rightarrow 7546 \rightarrow 14003 \rightarrow 44044
\] \[
a+b=17 \rightarrow 187 \rightarrow 968 \rightarrow 1837 \rightarrow 9218 \rightarrow 17347 \rightarrow 91718 \rightarrow \ldots
\] \[
a+b=18 \rightarrow 198 \rightarrow 1089 \rightarrow 10890 \rightarrow 20691 \rightarrow 40293 \rightarrow 79497
\] The only two values of $a+b$ which require exactly six steps are $a+b=16$ and $a+b=18$. However, the only $n$ for which $a+b=18$ is $n=99$, a palindrome. We are left with $97+79=\boxed{176}$, as we exclude the palindrome $n=88$.